Axiom of infinity
Axiom guaranteeing existence of an infinite set.
Wikipedia / Wikimedia Commons
The axiom of infinity is one of the axioms of Zermelo–Fraenkel set theory, first published by Ernst Zermelo as part of his set theory in 1908. It guarantees the existence of at least one infinite set, namely a set containing the natural numbers. In axiomatic set theory and the branches of mathematics and philosophy that use it, this axiom ensures that there is a set that includes the empty set and is closed under the successor operation, thus providing a foundation for the natural numbers.
- field
- Axiomatic set theory, mathematics, philosophy
- first_published_by
- Ernst Zermelo
- year_first_published
- 1908
- part_of
- Zermelo–Fraenkel set theory, von Neumann–Bernays–Gödel axioms
- guarantees
- Existence of at least one infinite set containing the natural numbers
Lore & Background
The axiom of infinity asserts the existence of a set I that contains the empty set and is closed under the successor operation, where the successor of x is defined as x ∪ {x}. This construction is closely related to the von Neumann construction of the natural numbers, in which zero is the empty set, one is {0}, two is {0,1}, and so on. A consequence of this definition is that every natural number is equal to the set of all preceding natural numbers.
Reader's Guide
The axiom of infinity is essential for the existence of the set of all natural numbers within Zermelo–Fraenkel set theory, as the other axioms are insufficient to prove the existence of such a set. By asserting that there is an inductive set containing the empty set and closed under successors, it provides the foundation for the natural numbers. The natural numbers can then be extracted from this infinite set using the axiom schema of specification, yielding a unique set N. This axiom is also one of the von Neumann–Bernays–Gödel axioms. Its significance lies in establishing the existence of infinite sets, which is fundamental for much of modern mathematics, including analysis and number theory.
Did You Know?
- The axiom of infinity was first published by Ernst Zermelo as part of his set theory in 1908.
- It guarantees the existence of at least one infinite set, namely a set containing the natural numbers.
- The axiom is closely related to the von Neumann construction of the natural numbers, where the successor of x is defined as x ∪ {x}.
- The natural numbers can be extracted from the infinite set using the axiom schema of specification.
Frequently Asked Questions
Who is the Axiom of Infinity?
It is a foundational principle within Zermelo–Fraenkel set theory, originally introduced by Ernst Zermelo in his 1908 axiomatization. In the 'cast list' of set-theoretic axioms, it is the one whose sole job is to declare that an infinite set actually exists.
What are the Axiom of Infinity's powers or role?
It guarantees the existence of a set that contains the empty set and is closed under the successor operation, meaning you can keep adding 'one more' forever. Concretely, it hands the system a set large enough to serve as the natural numbers.
How does the Axiom of Infinity's story end?
There is no dramatic finale; it simply sits in the background enabling virtually all of standard mathematics. Once accepted, arithmetic, real analysis, and most modern mathematical structures become formally available to build upon.
Why is the Axiom of Infinity important?
Without it, Zermelo–Fraenkel set theory can only reason about finite collections, which is far too weak for most mathematical work. It is the single axiom that opens the door to infinite structures and the number system we rely on daily.
Where does the Axiom of Infinity appear in the canon?
It is a core member of both Zermelo–Fraenkel set theory and the von Neumann–Bernays–Gödel class-theoretic axioms. So whether you are working in the smaller ZF framework or the broader NBG system, it is present doing its essential job.
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